1868.] Mr. J. C. Maxwell on Governors. 



279 



I have not succeeded in determining completely the conditions of sta- 

 bility of the motion from this equation ; but I have found two necessary 

 conditions, which are in fact the conditions of stability of the two governors 

 taken separately. If we write the equation 



-{-pn^-\-qn^-^rrn^-{-sn-\-t,- ' ...... (16) 



then, in order that the possible parts of all the roots shall be negative, it 

 is necessary that 



^g^-r and 23<5>^ (1/) 



I am not able to show that these conditions are sufficient. This com- 

 pound governor has been constructed and used. 



On the Motion of a Liquid iti a Tube revolvi^ig about a Vertical Axis. 



Mr. C. W. Siemens' s Liquid Governor. — -Let p be the density of the fluid, 

 k the section of the tube at a point whose distance from the origin mea- 

 sured along the tube is s, r, 8, z the coordinates of this point referred to 

 axes fixed with respect to the tube, Q the volume of hquid which passes 

 through any section in unit of time. Also let the following integrals, 

 taken over the whole tube, be 



fpkr^ds = k, fpi-dd = B, fp^ds = C, (1) 



the lower end of the tube being in the axis of motion. 



Let be the angle of position of the tube about the vertical axis, then 

 the m.oraent of momentum of the liquid in the tube is 



^i=^l+BQ (2) 



The moment of momentum of the liquid thrown out of the tube in unit 

 of time is 



dH' or^dd) , , 



— =p,.-Qj+p^-Q-cosc„ (3) 



where r is the radius at the orifice, k its section, and a the angle between 

 the direction of the tube there and the direction of motion. 

 The energy of motion of the fluid in the tube is 



+ BQg + iCQ"- (4) 



The energy of the fluid which escapes in unit of time is 



The work done by- the prime mover in turning the shaft in unit of 

 time is 



d^_d^ fdR dJTx 

 dt^dt \ dt^ dt ) 

 The work spent on the liquid in unit of time is 



f/W dW 



"dt^ dt ' 



cii 



